Demand responsive bus scheduling method considering stochastic road network and passenger spatio-temporal flexibility

Abstract

The application discloses a demand response bus scheduling method considering random road network and passenger space-time flexibility, and comprises the following steps: defining a demand response bus scheduling problem in a space-time network; establishing a vehicle flow space-time network and a passenger flow space-time network; verifying passenger flexibility; setting the cost of arcs in the vehicle flow space-time network and the passenger flow space-time network; setting a set of boarding and alighting nodes; constructing a demand response bus scheduling optimization model; and obtaining a result by solving the model. The application makes full use of the time and space flexibility of passengers, and is helpful to realize the effects of aggregating and dispersing passenger flow and reducing cost and increasing efficiency. Meanwhile, the random road network is considered in the scheduling optimization process, so that the application is closer to the actual situation. The application has a strong practical application prospect.

Description

Technical Field

[0001] This invention relates to the field of public transport scheduling, and more specifically to a demand-response public transport scheduling method that takes into account random road networks and passenger spatiotemporal flexibility. Background Technology

[0002] With the advancement of technology and the continuous improvement of people's living standards, urban public transportation systems have developed rapidly. At the same time, efficient and personalized travel services are increasingly favored by the public. Limited traditional public transportation networks struggle to meet the public's demands for high-quality, convenient, and comfortable travel services. Subsequently, a more flexible demand-responsive bus service was launched, where stops, timetables, and routes are determined by user needs. Demand-responsive buses are a new public transportation model between taxis and regular buses. This model matches groups with shared travel needs through an information platform, providing "one seat per person, fast and direct" bus sharing services. Passengers can participate in determining the origin and travel time, balancing personalized needs with efficient resource allocation, making it an important means of realizing mobility as a service and demand-responsive transportation.

[0003] However, its development still faces some challenges. One is the issue of demand dispersion. Compared to conventional public transport, it requires more capacity and human resources to meet the dispersed passenger flow. Therefore, how to aggregate more dispersed passenger flow to share costs and transport more passengers with fewer vehicles is an important question. Furthermore, in uncertain environments, the normal execution of vehicle transport tasks is easily disrupted, and travel times fluctuate. Since demand-response public transport has high requirements for operational reliability, it is necessary to consider the impact of stochastic road networks.

[0004] Traditional dispatching methods, which focus solely on improving vehicle control, often have limited potential for efficiency gains, thus restricting the release of dispatching efficiency. However, exploring potential efficiency improvements by considering passengers' willingness to pay and behavioral flexibility, and organically integrating this with demand-response bus dispatching solutions, can help aggregate dispersed passenger flow, reduce costs, increase efficiency, and achieve a win-win situation for both users and operators. Passengers possess mobility and flexibility of choice; they often accept "fine-tuning" of their travel plans within their reachable time and space, based on their willingness to pay. Bus operators can fully utilize passengers' time and spatial flexibility to adjust the distribution of travel demand, achieving refined reservation management and more efficient and cost-effective supply-demand matching, responding to and adjusting to different passenger needs.

[0005] Demand-response buses operate in open road environments, making them susceptible to interference from uncertain factors that can affect the connection between bus routes and tasks. Therefore, considering random road networks during the scheduling optimization process will be more in line with reality.

[0006] Bus dispatching primarily aims to optimize vehicle operation by allocating tasks according to a pre-defined timetable, ensuring clear and consistent operational status for each vehicle. For demand-response buses, where timetables and routes constantly change based on user demand, the demands on dispatching are even higher. The quality of the dispatch plan significantly impacts overall operating costs and is a primary consideration for bus companies. Furthermore, accurate and effective timetable design and optimization are crucial for improving service levels. Optimized dispatching of demand-response buses is vital for operational success. Summary of the Invention

[0007] In order to overcome the shortcomings and deficiencies of the existing technology, the present invention provides a demand-response bus scheduling method that takes into account random road networks and passenger spatiotemporal flexibility.

[0008] The present invention adopts the following technical solution:

[0009] A demand-response bus scheduling method that considers stochastic road networks and passenger spatiotemporal flexibility includes:

[0010] Constructing a spatiotemporal network for vehicle flow and a spatiotemporal network for passenger flow;

[0011] Passenger flexibility verification;

[0012] The cost of setting up arcs in the spatiotemporal networks of vehicle flow and passenger flow;

[0013] Set the set of boarding and alighting nodes;

[0014] A demand-response bus scheduling optimization model is constructed and solved to obtain the demand-response bus scheduling optimization scheme.

[0015] Furthermore, including:

[0016] The traffic flow spatiotemporal network corresponds to a demand-response bus and reflects the driving status of each bus;

[0017] There are two axes: the horizontal axis represents the spatial dimension, indicating spatial location, specifically a station; the vertical axis represents the time dimension, indicating the time corresponding to a certain event.

[0018] Nodes in the traffic flow spatiotemporal network have temporal and spatial attributes, representing the location of a bus at a certain time.

[0019] In the spatiotemporal network of traffic flow, an arc represents the trajectory of a vehicle.

[0020] Furthermore, the arcs of the vehicle spatiotemporal network include four types, namely:

[0021] Random vehicle travel arc: represents the journey of a bus from one station to another, which corresponds to a journey time. The corresponding arc cost is the variable cost of the vehicle performing the transportation task plus the penalty cost that reflects the possible early or late arrival in actual operation rather than the trip plan.

[0022] Vehicle waiting arc: represents the time a vehicle spends waiting at a certain station during a certain period. The corresponding arc cost is set as the vehicle's waiting time multiplied by the driver's cost per unit of time.

[0023] Vehicle departure arc: This arc connects the depot and the station. The departure end is the depot, and the destination end is the passenger boarding station. The corresponding arc cost represents the operating cost of arranging vehicle departure.

[0024] Vehicle return arc: This refers to the vehicle returning to the depot after serving the last passenger and reaching the last drop-off point. The arc cost of the vehicle return arc is set to 0.

[0025] Furthermore, the passenger flow spatiotemporal network describes passenger flow patterns, with each layer of the passenger flow spatiotemporal network corresponding to a passenger demand with a start-destination-time sequence.

[0026] The spatiotemporal network diagram of passenger flow has two axes: the vertical axis represents the time dimension, that is, the time corresponding to a certain event; the horizontal axis represents the spatial dimension, that is, the spatial location.

[0027] In the spatiotemporal network of passenger flow, nodes and arcs are set up. Nodes have time and space attributes, and arcs represent the movement of passengers. Arcs correspond to the starting node and the destination node.

[0028] The duration of the passenger flow spatiotemporal network represents the travel time window allowed by the relevant origin-destination-time requirements.

[0029] Furthermore, the passenger flow spatiotemporal network includes four types of arcs, namely:

[0030] Passenger random travel arc: The passenger random travel arc represents the passenger journey from one station to another, which corresponds to a journey time. The number of random travel arcs in each passenger flow spatiotemporal network corresponds to the number of vehicle random travel arcs in the vehicle flow spatiotemporal network. Its arc cost is specifically the variable cost of serving passenger travel plus the penalty cost that reflects the possible early or late arrival in actual operation rather than the trip plan.

[0031] Passenger waiting arc: This represents the time a passenger spends waiting at a station during a certain period. The cost of the passenger waiting arc is set as the passenger's waiting time multiplied by the cost per unit time the passenger spends on the train.

[0032] Passenger Virtual Departure Arc: Connects the virtual depot and the station. The departure point is the virtual depot, and the destination point is the passenger boarding station. Its corresponding arc cost is set to 0.

[0033] Passenger Virtual Return Arc: This represents the return of a passenger to the virtual depot after arriving at their destination station, and its corresponding arc cost is 0.

[0034] Furthermore, the passenger flexibility verification is reflected by the amount that passengers are willing to pay for different levels of flexibility in time and space.

[0035] Furthermore, the process for obtaining the amount of money the person is willing to pay is as follows:

[0036] A virtual market environment is established, and the selection experiment method in non-market valuation is adopted. In the virtual market environment, the public's willingness to pay for different departure times and departure stations, i.e., different degrees of time and space flexibility, is inquired. An appropriate econometric model is then established to obtain the amount that passengers are willing to pay for different degrees of time and space flexibility.

[0037] Furthermore, the method for setting the cost of arcs in the vehicle flow spatiotemporal network and passenger flow spatiotemporal network is as follows:

[0038] The arc cost of the spatiotemporal networks of vehicle flow and passenger flow is respectively represented by the following piecewise functions:

[0039]

[0040]

[0041] Furthermore, the setting of the boarding and alighting node set is extracted from the passenger flow spatiotemporal network, specifically as follows:

[0042] Set of boarding nodes:

[0043] The set of boarding nodes is created according to the following formula, where Describing the set-fetching function for the boarding node

[0044]

[0045] It is the first The order data in the layered passenger flow spatiotemporal network is generated from passenger demand. N Stop It refers to the number of stations. This is the maximum service time.

[0046] Set of disembarkation nodes:

[0047] The set of disembarkation nodes is created using the following formula.

[0048]

[0049] in This represents the set-fetching function for the disembarkation node. Representative station direct travel time matrix, This represents the delay factor.

[0050] Furthermore, the order data includes multiple passenger boarding stations, multiple passenger alighting stations, ideal boarding time, number of passengers, and time flexibility parameters. The first boarding point is the ideal boarding station, and the first alighting point is the ideal alighting station.

[0051] Furthermore, the method for constructing the demand response bus scheduling optimization model is as follows:

[0052] The objective function minimizes the total operating cost, which includes the operating cost of the demand-response bus, the travel costs of all passengers on the bus, and penalties for unserved passengers.

[0053]

[0054] Vehicle flow spatiotemporal network parking lot constraints: Formulas (2) and (3) indicate that the number of vehicles leaving and returning to the parking lot is the same in each vehicle flow spatiotemporal network.

[0055]

[0056]

[0057] The flow conservation constraint of the traffic flow spatiotemporal network nodes, as shown in formula (4), represents the flow conservation of each node in the traffic flow spatiotemporal network. This constraint is to ensure the consistency of the flow when entering and leaving the traffic flow spatiotemporal network nodes. That is, in the station corresponding to the traffic flow spatiotemporal network, after the demand response bus arrives at the station from other stations, the next step is to continue to perform the task or return to the depot.

[0058]

[0059] Passenger flow spatiotemporal network supply and demand node flow conservation constraint:

[0060]

[0061]

[0062] Passenger flow spatiotemporal network node flow conservation constraint. Equation (7) represents the flow conservation of each node in each passenger flow spatiotemporal network. This constraint is to ensure the consistency of flow between inbound and outbound passenger flow spatiotemporal network nodes.

[0063]

[0064] Passenger number constraint. Formula (8) shows that the sum of passenger travel arcs multiplied by the number of passengers in the passenger flow spatiotemporal network should be less than or equal to the sum of the corresponding vehicle travel arcs multiplied by the vehicle capacity in all vehicle flow spatiotemporal networks. This constraint ensures that passengers can board a vehicle, because if a passenger can travel from node i to node j, there must be a vehicle to transport them. For a vehicle, it can only serve the corresponding passenger demand if it travels from node i to node j. The set of all arcs in the vehicle or passenger flow spatiotemporal network. In the diagram, arc ij represents the journey from node i to node j. Arc (i,j) connects the demand-response bus and the passenger, ensuring that the passenger boards the demand-response bus from node i to node j.

[0065]

[0066] Decision variable constraints, as specified in formulas (9)-(12), stipulate that all decision variables in the vehicle flow spatiotemporal network and passenger flow spatiotemporal network are either 0 or 1:

[0067]

[0068]

[0069]

[0070]

[0071] Furthermore, the constructed model is a mixed-integer linear programming model.

[0072] The beneficial effects of this invention are:

[0073] This invention takes into account random road networks and passenger spatiotemporal flexibility, considers the impact of uncertainties in open road environments, and fully explores the potential for improving scheduling efficiency through time and space flexibility from the passenger's perspective. This makes the proposed method more in line with reality and more cost-effective, providing a more efficient scheduling solution for demand-response bus operations, thus achieving the effect of cost reduction and efficiency improvement. Attached Figure Description

[0074] Figure 1 This is a flowchart of the method of the present invention;

[0075] Figures 2(a)-2(d) It is a schematic diagram of the impact on passenger time and space flexibility and bus scheduling, including service network topology, train schedules without time and space adjustment, time adjustment and space adjustment;

[0076] Figure 3 It is a diagram showing the transformation between physical transportation networks and spatiotemporal networks;

[0077] Figure 4 This is a schematic diagram of the spatiotemporal network structure of traffic flow;

[0078] Figure 5 This is a diagram illustrating the penalty costs incurred during the vehicle's journey;

[0079] Figure 6 This is a schematic diagram of the spatiotemporal network of passenger flow;

[0080] Figure 7 This is a diagram illustrating the penalty costs incurred by passengers during their travels.

[0081] Figure 8 This is a schematic diagram of the questionnaire;

[0082] Figure 9 This is a schematic diagram illustrating the cost calculation for spatiotemporal flexibility.

[0083] Figures 10(a)-10(b) This is a diagram illustrating the lag factor;

[0084] Figure 11 This is a diagram illustrating the acquisition of the boarding node set;

[0085] Figure 12 This is a diagram illustrating the process of obtaining the set of disembarkation nodes (taking a portion of the boarding nodes as an example).

[0086] Figure 13 This is a schematic diagram of the Sioux Falls network structure;

[0087] Figures 14(a)-14(b) These are schematic diagrams illustrating vehicle scheduling scenarios with and without spatiotemporal flexibility.

[0088] Figures 15(a)-15(c) The comparisons are between those with and without time and space flexibility, including cost structure, number of service orders, and offset comparison diagrams;

[0089] Figures 16(a)-16(b) These are diagrams illustrating how the cost structure and fleet size change flexibly over different times.

[0090] Figures 17(a)-17(b) These are schematic diagrams illustrating how cost structure and fleet size vary with different spatial flexibility.

[0091] Figures 18(a)-18(b) These are schematic diagrams illustrating system performance metrics under different time flexibility conditions, namely, full spatial flexibility and no order rejection mode. Detailed Implementation

[0092] The present invention will be further described in detail below with reference to the embodiments and accompanying drawings, but the embodiments of the present invention are not limited thereto.

[0093] Example

[0094] like Figure 1As shown, a demand-response bus scheduling method considering stochastic road networks and passenger spatiotemporal flexibility includes the following steps:

[0095] Establish a spatiotemporal network for vehicle flow and a spatiotemporal network for passenger flow.

[0096] Passenger flexibility verification.

[0097] The cost of setting up arcs in the spatiotemporal networks of vehicle flow and passenger flow.

[0098] Set up a set of boarding and alighting nodes.

[0099] Construct a demand-response bus scheduling optimization model.

[0100] The solution yields the result.

[0101] The specific process of the bus dispatching method is as follows:

[0102] 1. The conceptual framework and construction process of this method:

[0103] Demand-response bus dispatching aims to allocate vehicles to fulfill various user demands. This involves optimizing vehicle routes to ensure vehicles arrive at pre-booked stations in the correct order, providing clear information on each vehicle's operation for optimal performance. Since demand-response bus timetables and routes constantly change based on user needs, the demands on dispatching are higher. The quality of the dispatch plan significantly impacts overall operating costs and is a primary consideration for bus companies. Furthermore, accurate and effective timetable design and optimization are crucial for improving service levels. Optimized dispatching of demand-response buses is vital for operational success.

[0104] Passenger spatiotemporal flexibility plays a crucial role. To help clarify the issue of fully leveraging passenger spatiotemporal flexibility in this patent, a simple service network is used as an example to illustrate how these two types of flexibility can be used to reduce the fleet size required to meet travel demand, as shown in Figure 2(a). This network consists of one depot and four alternative stations. As shown in Figure 2(b), assuming there are two pre-booked travel demands, one from station 1 to station 2 and the other from station 3 to station 4, the system plans vehicle routes based on the collected pre-booked departure times and stations. When passenger spatiotemporal flexibility is not available, two vehicles are needed to complete the task. As shown in Figure 2(c), if passengers at station 1 are willing to depart 5 minutes earlier, the originally planned bus routes 1 and 2 can be combined into one vehicle. As shown in Figure 2(d), if passengers are willing to change their boarding location from station 2 to station 3, the originally planned two bus routes can also be combined into one vehicle, shortening the journey of bus route 2 (Figure 2(a)). It can be observed that appropriate time and space adjustments can transform infeasible train connections into feasible train chains, thereby saving capacity and increasing load factor. Therefore, public transport operators can leverage passenger time and space flexibility to reduce operating costs and improve efficiency.

[0105] In the above introduction, we utilized passengers' spatial and temporal flexibility. Therefore, we need to implement incentives to encourage users to "fine-tune" their departure location and time. This cost can be viewed as a discount on the user's spatial and temporal flexibility. This cost is related to some "penalty parameters," namely, the penalty for departing earlier (wtp). e Penalty for delayed departure time Spatial transfer distance penalty wtp d These parameters were obtained through a willingness-to-pay survey, which will be introduced later.

[0106] This invention considers the spatiotemporal flexibility of passengers in a random road network and provides a solution for demand-response bus scheduling.

[0107] For ease of reading, a symbol table is provided below:

[0108] Table 1 Symbol Table

[0109]

[0110]

[0111] *The vehicle / passenger flow network here is an abbreviation for the vehicle / passenger flow spatiotemporal network.

[0112] 2. Basic Assumptions

[0113] (1) In the demand response bus scheduling model, demand response buses depart from the same depot and return to the same depot after completing all tasks.

[0114] (2) All passenger demand is known and served by a fixed-size fleet.

[0115] (3) Passengers can choose which orders to accept, avoiding the over-allocation of fleet resources.

[0116] (4) The travel time follows a specific probability distribution.

[0117] (5) Once a group of passengers have placed an order, they can only board and disembark at the same time and cannot be split into multiple vehicles for transportation.

[0118] 3. The establishment of the vehicle flow spatiotemporal network and passenger flow spatiotemporal network mainly involves defining the demand response public transport scheduling problem within the spatiotemporal network.

[0119] The demand-response bus scheduling problem can be abstracted as a graph theory model. We use spatiotemporal network modeling techniques to construct the route and timetable compilation model for demand-response buses, considering the impact of random travel time and fully exploring the spatiotemporal flexibility of passengers. The spatiotemporal network illustrates the changes in the spatial location of vehicles / passengers over time. The trajectories of vehicles / passengers are represented by arcs in the network, which is intuitive and easy to model mathematically. We designed spatiotemporal networks for vehicle flow and passenger flow respectively.

[0120] Figure 3 This demonstrates the transformation between a physical transportation network and a spatiotemporal network. The physical transportation network has three stations: the shortest travel time from station 1 to station 2 is 15 minutes, the shortest travel time from station 1 to station 3 is 20 minutes, and the shortest travel time from station 2 to station 3 is 15 minutes. Considering a spatiotemporal network model with a time span of 30 minutes and a network interval of 5 minutes, the transformed network is as follows. Figure 3 As shown on the right. The diagram also illustrates the direct routes between the various stations.

[0121] The specific process for establishing the spatiotemporal networks for vehicle flow and passenger flow is as follows:

[0122] (1) Spatiotemporal network of traffic flow

[0123] Considering random travel time, the potential operation of a vehicle within a certain time and space is as follows: Figure 4 As shown in the diagram, each traffic flow spatiotemporal network corresponds to a demand-response bus, and the movement of each bus is represented by the corresponding spatiotemporal network diagram. The spatiotemporal network diagram has two axes: the vertical axis represents the time dimension, indicating the time corresponding to an event; the horizontal axis represents the spatial dimension, indicating spatial location. There are two important elements in the spatiotemporal network diagram: nodes and arcs. Nodes have two attributes: temporal and spatial. Therefore, a node can represent a location at a specific time. Arcs represent the movement of vehicles, corresponding to the starting node and the destination node. Because nodes have spatiotemporal attributes, the information about vehicle movement can be richly represented, i.e., when and where the vehicle travels from and to when and where. A complete set of arcs in a spatiotemporal network diagram can express the specific process of a vehicle performing a task. The vehicle's trajectory and travel time can be easily obtained from the diagram. This is the advantage of spatiotemporal network diagrams; they can simplify problems and greatly aid in problem-solving.

[0124] There are four types of arcs in total.

[0125] Vehicle random driving arc

[0126] A vehicle's random travel arc represents a vehicle's journey from one station to another, corresponding to a travel time. The traffic flow spatiotemporal network contains all vehicle random travel arcs, and each journey between stations has several possible travel times. The arc flow in the traffic flow spatiotemporal network is also a binary variable, representing whether the journey associated with the random travel time is executed; 1 represents execution, and 0 represents no execution. Arc cost is the variable cost of a vehicle performing a transportation task plus the penalty cost reflecting adjustments made in actual operation to possible early or late arrivals rather than the planned journey (discussed below).

[0127] In our problem, we consider the random travel time of vehicle journeys by setting several possible random travel arcs for a single vehicle journey. Each random travel arc corresponds to one possible scenario in the journey, i.e., a travel time and a probability. That is, for a journey between stations, there are multiple possible travel times for a vehicle, each travel time corresponding to a probability, and the random travel arc is a combination of travel time and probability. The purpose of designing random travel arcs is to determine the appropriate travel time for each vehicle journey in the planned timetable. During operation, the planned travel time is not necessarily equal to the actual travel time. If the random travel time of the vehicle during transportation differs from the planned travel time, it indicates that the vehicle's travel time is inconsistent with the planned time. Below, we will use examples to explain this problem. Figure 5 As shown, assuming there are four possible travel times for a vehicle journey, namely 10, 15, 20, and 25 minutes, with corresponding probabilities of 0.25, 0.35, 0.30, and 0.10. Comparing the four possible vehicle travel arcs, we can see that the times of x1, x2, and x4 are inconsistent with the time of x3, differing by 10, 5, and 5 minutes respectively.

[0128] No. Train numbers in the spatiotemporal network diagram of layered traffic flow<i,j> Planned travel time The total penalty cost is calculated as follows:

[0129]

[0130] In the above formula, It is the first Train numbers in the spatiotemporal network diagram of layered traffic flow<i,j> Planned travel time The total penalty. It is the first Train numbers in the spatiotemporal network diagram of layered traffic flow<i,j> Early or late arrival scenarios and planned travel time The difference, It's important to note that if the vehicle arrives early (meaning the travel time s is less than the planned travel time),... ),So If the vehicle is late (meaning the journey time s is longer than the planned journey time), ),So Train number<i,j> Early / late arrivals are penalized per unit of time. For passengers, arriving early or late deviates from their ideal travel time, therefore, a penalty is imposed for early or late arrivals. i,j Train number<i,j> Adjustment coefficient for the impact of lateness on subsequent trains. When a train's journey time exceeds the scheduled journey time, the impact of the delay is not fully transmitted to downstream trains, due to factors such as driver recovery and buffer time. Therefore, a coefficient ρ with a value less than 1 is set. i,j This addresses the propagation of vehicle delays. The earlier a delay occurs, the smaller the impact, because there is more buffer time and opportunity for driver recovery. Therefore, ρ i,j It may vary over time. η i,j Train number<i,j> The average number of people boarding the vehicle downstream is taken as the average number of people per order. This is a penalty for passengers arriving early or late per unit of time. For passengers, arriving early or late deviates from their ideal travel time, therefore a penalty is imposed for early or late arrivals. ε i,j Train number<i,j> The adjustment factor for the impact of lateness on subsequent passengers. This value is greater than 1; the later the delay, the greater the impact, because the buffer time decreases and the downstream passenger backlog increases. Therefore, ε i,j It may vary over time.

[0131] For example, such as Figure 5 As shown, assuming in train number<i,j> The probabilities of all possible random vehicle travel arcs are 0.25, 0.35, 0.30, and 0.10. Comparing the travel times of all possible random vehicle travel arcs, the time differences between vehicle travel arcs x1, x2, and x4 relative to x3 are 10, 5, and 5 minutes, respectively. The penalty cost is calculated as follows:

[0132] Vehicle waiting arc

[0133] A vehicle waiting arc represents the time a vehicle spends waiting at a station during a given period. During this time, the vehicle is not performing any transport tasks and remains stationary, waiting to proceed to the next station. The cost of a vehicle waiting arc is set as the vehicle's waiting time multiplied by the driver's cost per unit of time. The upper bound of the vehicle waiting arc flow is 1, indicating that at most one vehicle is waiting at the station during this period; the lower bound is 0, meaning that no vehicle is waiting at the station during this period.

[0134] Vehicle exit arc

[0135] The vehicle departure arc connects the depot and the station, with the depot as the departure point and the passenger boarding station as the destination. Vehicle departure is determined by the passenger's chosen boarding location, indicating the arrival time at which station. The cost of the vehicle departure arc represents the operational cost of scheduling vehicle departures. The upper bound of the vehicle departure arc flow is 1, meaning the vehicle departs from the depot and arrives at the station; the lower bound is 0, meaning the vehicle does not depart from the depot and arrives at the station.

[0136] Vehicle retraction arc

[0137] To ensure flow conservation in the spatiotemporal network of vehicle traffic, we employ a vehicle return arc, connecting the vehicle to the depot from the last drop-off station. This represents the vehicle returning to the depot after serving the last passenger and reaching the last drop-off station. The arc cost of the vehicle return arc is set to 0. The upper bound of the arc flow is 1, meaning the vehicle returns to the depot after completing all tasks; the lower bound of the arc flow is 0, meaning the vehicle does not return to the depot from the station.

[0138] (2) Spatiotemporal network of passenger flow

[0139] To facilitate problem-solving, we set up the passenger flow spatiotemporal network to resemble the vehicle flow spatiotemporal network, such as... Figure 6 As shown, the passenger flow spatiotemporal network describes the flow of passengers and displays the passenger transport situation of vehicles. Each layer of the passenger flow spatiotemporal network corresponds to a passenger demand from origin to destination to time, which is related to the requirements of passenger reservation demand response to public transportation. The travel situation of the same batch of passengers is expressed by the corresponding spatiotemporal network diagram. Similarly, there are two axes in the passenger flow spatiotemporal network diagram: the vertical axis represents the time dimension, that is, the time corresponding to a certain event; the horizontal axis represents the spatial dimension, that is, the spatial location. Two important elements in the spatiotemporal network are nodes and arcs, which will be introduced in detail below. Nodes have two attributes: time attribute and spatial attribute. Therefore, a node can represent a certain location at a certain time. Arcs can represent the movement of passengers. Arcs correspond to the origin node and the destination node. Since nodes have spatiotemporal attributes, the information of passenger movement can also be reflected very richly, that is, when and where passengers board the demand response bus from and to where. A complete set of arcs in a spatiotemporal network diagram can express the specific process of a batch of passenger transport. The travel trajectory of passengers on the vehicle and the travel time can be easily obtained from the diagram.

[0140] The duration of a passenger flow spatiotemporal network represents the travel time window allowed by the relevant origin-destination-time demand (ODT). The duration of the network varies depending on different passenger demands; therefore, different passenger flow spatiotemporal networks may have different durations.

[0141] There are four types of arcs in the spatiotemporal network of passenger flow.

[0142] Passenger random travel arc

[0143] We determine the appropriate travel time for each trip in the planned timetable by designing passenger random travel arcs. The number of passenger random travel arcs in each passenger flow spatiotemporal network corresponds to the number of vehicle random travel arcs in the vehicle flow spatiotemporal network. A passenger random travel arc represents a passenger trip from one station to another, corresponding to one travel time. The passenger flow spatiotemporal network contains a group of passenger (i.e., the same origin-destination-time demand) random travel arcs, and trips between stations have several possible travel times. The arc flow in the passenger flow spatiotemporal network is also a binary variable, representing whether the trip associated with the random travel time has taken place; 1 represents a trip, and 0 represents no trip. The arc cost is the variable cost of serving passenger trips plus the penalty cost reflecting adjustments made in actual operation to possible early or late arrivals rather than the trip plan (discussed below).

[0144] In our problem, we consider the random travel time of passengers by setting several possible random travel arcs for a single passenger trip. Each random travel arc corresponds to one possible scenario in the trip, i.e., a travel time and a probability. That is, for trips between stations, there are multiple vehicle travel times, and therefore multiple passenger travel times. Each travel time corresponds to a probability, and the random travel arc is a combination of travel time and probability. The purpose of designing the random travel arc is to determine the appropriate travel time for each passenger trip in the planned timetable. In operation, the planned travel time is not necessarily equal to the actual travel time. If the random travel time of a passenger during the trip differs from the planned travel time, it means that the passenger's actual travel time is inconsistent with the planned time. Below, we will use examples to explain this problem. Figure 7 As shown, assuming there are four possible travel times for a vehicle trip, namely 10, 15, 20, and 25 minutes, with corresponding probabilities of 0.25, 0.35, 0.30, and 0.10. Comparing the four possible random travel arcs of passengers, we can see that the times of x1, x2, and x4 are inconsistent with the time of x3, differing by 10, 5, and 5 minutes respectively.

[0145] No. Train numbers in the spatiotemporal network diagram of passenger flow<i,j> Planned travel time The total penalty cost is calculated as follows:

[0146]

[0147] In the above formula, It is the first Train numbers in the spatiotemporal network diagram of passenger flow<i,j> Planned travel time The total penalty. ζ i,jIs it due to passengers being late for the train?<i,j> Adjustment factor for the impact on subsequent trains. When the arrival time exceeds the scheduled time, the impact of the delay will not be fully passed on to downstream trains, for example, through driving recovery. A coefficient ζ with a value less than 1 is set. i,j To address the propagation of vehicle delays. i,j It may vary over time.

[0148] For example, such as Figure 7 As shown, assuming in train number<i,j> The probabilities of all passenger random travel arcs are 0.25, 0.35, 0.30, and 0.10. Comparing the travel times of all possible passenger random travel arcs, the travel time differences for passenger random travel arcs x1, x2, and x4 relative to x3 are 10, 5, and 5 minutes, respectively. The penalty cost is calculated as follows:

[0149] Passenger waiting arc

[0150] A passenger waiting arc represents the time a passenger spends waiting at a station during a given period. During this time, the vehicle is not performing any transport tasks and remains stationary, waiting to proceed to the next station. The cost of the passenger waiting arc is set as the passenger's waiting time multiplied by the cost per unit of time the passenger spends on the vehicle. The upper bound of the passenger waiting arc flow is 1, indicating that at most one group of passengers remains at the station during this period; the lower bound is 0, meaning that no passengers remain at the station during this period.

[0151] Passenger virtual exit arc

[0152] The virtual departure arc connects the virtual depot and the station, with the virtual depot as the departure point and the passenger boarding station as the destination. Passenger flow is actually determined by the passenger's chosen boarding location, indicating when they will arrive at which station. The cost of the virtual departure arc is set to 0. The upper bound of the virtual departure arc flow is 1, meaning the passenger departs from the virtual depot and arrives at the station; the lower bound of the virtual departure arc flow is 0, meaning the passenger does not depart from the virtual depot and arrive at the station.

[0153] Passenger virtual closing arc

[0154] To ensure flow conservation in the passenger spatiotemporal network, we employ a virtual passenger return arc, connecting the destination station to the virtual depot. This represents a passenger returning to the virtual depot after reaching the destination station. The arc cost of the virtual passenger return arc is set to 0. The upper bound of the arc flow is 1, meaning the passenger returns to the virtual depot after reaching their destination; the lower bound of the arc flow is 0, meaning the passenger does not return to the virtual depot from the station.

[0155] 4. The passenger flexibility verification specifically includes:

[0156] (1) Definition

[0157] In our problem, the optimization process considered spatiotemporal flexibility, allowing adjustments to passengers' travel time and location, expanding their time and space window, enabling the bus company to achieve lower costs, and creating a win-win situation for both the bus company and passengers. The preceding description mentioned adjusting passenger travel time and location. When passengers book a demand-response bus, they submit their travel requests, at which point the ideal departure time and ideal stop need to be determined. Then, the demand-response bus company's dispatch center performs optimization calculations based on all collected passenger travel requests, resulting in the optimized departure time and stop for the passenger. However, the optimized travel time and stop may not be the ideal time and stop determined by the passenger at the time of booking. In such cases, fare reductions or preferential treatment such as complimentary ride coupons are necessary.

[0158] Therefore, we need to further explore the appropriate discount amounts corresponding to different levels of time and location flexibility. Considering the limitations of bus companies, which cannot represent the general public, we need to determine the discount amounts more precisely from the passenger's perspective. This would allow us to minimize the bus company's costs while maintaining a price acceptable to the general public, thus better aligning with users' actual needs and increasing user acceptance. To address this issue, we have introduced the concept of willingness to pay, focusing on the passenger's perspective to maximize the fulfillment of their payment needs.

[0159] Willingness to pay refers to the degree to which we are willing to pay for a product or service with certain characteristics. It reflects our level of interest in a particular good or service and is also the maximum price we are willing to pay. We introduced willingness to pay precisely to address the issue of varying levels of discounts depending on the degree of flexibility in terms of time and location.

[0160] The time and spatial flexibility of demand-response public transportation is not a market resource, and its economic value cannot be derived from normal supply and demand market mechanisms. Therefore, we use the choice experiment method, a non-market valuation method, to estimate the amount passengers are willing to pay for different levels of time and spatial flexibility. This method will be explained in detail later.

[0161] (2) Willingness to pay survey

[0162] A virtual market environment was established, employing the choice experiment method from non-market valuation techniques. This virtual environment was used to inquire about the public's willingness to pay for different departure times and departure points—that is, different levels of time and spatial flexibility. An appropriate econometric model was then developed to obtain the amount passengers were willing to pay for different levels of time and spatial flexibility. The primary reason for choosing the choice experiment method is that it starts from the perspective of a broad range of users wanting to use demand-responsive public transportation and quantifies user perception through their willingness to pay.

[0163] In our problem, the attributes to be studied have been determined: temporal and spatial flexibility. Specifically, this refers to the passenger's departure time, departure station, and destination station. The levels of temporal flexibility are: departure time 20, 15, 10, and 5 minutes earlier (ideal time); and departure time 20, 15, 10, and 5 minutes later (ideal time). The levels of spatial flexibility are: walking distance 0, 100, 150, 200, 250, and 300 meters.

[0164] If a full-scale experiment were used to collect data, it would require numerous experiments, which is impractical. Orthogonal experimental design, however, uses a standardized table (i.e., an orthogonal array) to select a representative subset of data from a full-scale experiment. In a sense, the full-scale experiment can be represented by this smaller subset. Therefore, we used the orthogonal experimental design to design the selection set, and the results are shown in Table 2.

[0165] Table 2 Selection Set

[0166]

[0167]

[0168] *Time is used here to indicate time flexibility; negative values ​​indicate earlier arrival and positive values ​​indicate later arrival. Unit: minutes.

[0169] **Here, distance is used to represent spatial flexibility, and numerical values ​​represent the extent of expansion. The unit is meters.**

[0170] We collected information through a questionnaire survey. The questionnaire consisted of two parts: the first part introduced demand-responsive public transportation; the second part included the content of the selection experiment method, namely the design of a combination of time and space flexibility attributes; and the third part was about users' demographic characteristics and socioeconomic status, mainly including gender, age, education level, and income level. Figure 8 This presents a case study of the second part of a questionnaire.

[0171] The demographic characteristics and socioeconomic status results are shown in Table 3.

[0172] Table 3. Results of the questionnaire on demographic characteristics and socioeconomic status.

[0173]

[0174]

[0175] Conditional logit model

[0176] We use a conditional logit model to estimate willingness to pay. We assume each user faces a set of alternative spatiotemporal flexibility attribute states. The spatiotemporal flexibility attribute state r can bring utility U to the user m who selects it. mr Therefore, we have:

[0177] U mr =V mr +ε mr (3)

[0178] And V mr =βX mr (m=1,2,3…|m|) (4)

[0179] V mr The feature vectors X of various spatiotemporal flexibility attributes mr The utility function consists of β, which is the estimated parameter, and ε. mr Other relevant influencing factors. Based on the principle of utility maximization when consumers make choices, the probability that a tourist will choose the spatiotemporal flexibility attribute state r is:

[0180]

[0181] Here, a logit model is used for parameter estimation. Assume the random error term ε... mr If the combinations are independent and follow an extreme value distribution, then the probability Pr that user m chooses the optimal combination r is... mr for:

[0182]

[0183] Equation (6) is the conditional logit model. Where V mr r and The meaning is the same as in equation (4), where θ is a scalar parameter, inversely proportional to the standard deviation of the random error term, and is usually taken as 1. Parameter estimation for the conditional logit model typically employs the maximum likelihood estimation method, and its log-likelihood function is:

[0184]

[0185] Among them, y mrIt is an indicator variable, representing a value of 1 when the m-th person selects the r-th candidate attribute set, and 0 otherwise. Therefore, for each m, there is one and only one y. mr =1; L is the maximum likelihood function. When the state of a certain spatiotemporal flexibility attribute changes, the willingness to pay of the respondent m can be obtained by formula (8):

[0186]

[0187] In the formula and These represent the current state of the attribute and its changed state, respectively; β P Let be the coefficient of the monetary attribute. Then the marginal willing payoff for a certain spatiotemporal flexibility attribute is:

[0188] MWTP=-β F / β P (9)

[0189] Where β F This represents the spatiotemporal flexibility attribute coefficient.

[0190] Results Analysis

[0191] Table 4. Parameter estimation for the conditional logic model

[0192]

[0193] As shown in Table 4, the p-value is very small, indicating a good fit. Based on the above results, we can conclude that both walking distance and time adjustments have negative effects, which aligns with reality. This means that making passengers walk more or adjust their departure time incurs a cost. Furthermore, in terms of time, making passengers depart later costs more than making them depart earlier, which is also very realistic, as we are all sensitive to being late.

[0194] The results show that there are indeed issues with passengers' willingness to pay for flexibility in terms of time and space in some cases. By using incentives, passengers can accept changes in station location or travel time, which can greatly help improve scheduling efficiency.

[0195] Passenger flexibility costs can be derived from willingness to pay, which is calculated as the ratio of the spatiotemporal flexibility coefficient to the monetary attribute coefficient. Based on the above results, the unit time flexibility cost (time advance), unit time flexibility cost (time delay), and unit spatial flexibility cost are respectively: wtp e =0.2694¥ / min, wtp d = 0.0211￥ / m.

[0196] 5. The cost of the arc in the aforementioned vehicle flow spatiotemporal network and passenger flow spatiotemporal network is as follows:

[0197] In our model, system costs consist of operating costs and user costs. Operating costs include fixed vehicle costs and variable costs. Regarding variable costs, we consider two components (time and space). One is driver labor costs, and the other includes fuel consumption costs, lubricant, tire, and maintenance costs, etc. Driver labor costs depend on the working hours, i.e., the time taken from leaving the depot to returning. The others depend on the distance traveled.

[0198] We incorporate various costs into arc costs. For a typical arc, the arc cost is the product of the time difference between nodes and the value of a unit of time. Other costs are handled as follows: a) Fixed vehicle costs can be simply added to the departure arc; b) Random trip arc costs are the costs of all random trip arcs, each corresponding to a trip time and probability, and the cost is calculated in the planned trip arc; c) Flexibility costs, to ensure passengers accept "fine-tuning" of their travel time and space, require compensation through fare reductions. However, fare reductions increase costs for the bus company, so this cost is added to the cost of the arc corresponding to earlier or later departure times or changes in stops.

[0199] In the passenger flow spatiotemporal network, each layer corresponds to a passenger order, and the arc flow represents the trajectory of the passenger for that order. Therefore, we incorporate spatiotemporal flexibility costs into the passenger flow spatiotemporal network to facilitate model construction and solution. Specifically, we add this cost to the first arc cost of the arc flow corresponding to the passenger order (considering departure time and station offset), expressing the penalty for different departure times and stations (which may deviate from the passenger's ideal departure time and station). For example, such as... Figure 9 As shown, suppose a passenger wants to board at station 1 at 06:05, allowing a 5-minute time offset to board at station 2, and wants to alight at station 4. If the passenger's ideal time and station are followed, the sequence is (virtual departure arc - arc 3 - arc 8 - virtual arrival arc), where arc 3 incurs no flexibility cost. However, if the optimization result requires adjusting the passenger's departure time or location, the cost is added to the first arc segment of the passenger's journey, depending on the scenario. For example, if the optimization result requires the passenger to board at station 2 at 6:00, the sequence is (virtual departure arc - arc 2 - arc 7 - virtual arrival arc), where arc 2's cost includes both time and location flexibility costs (due to both time and location offsets).

[0200] The flexibility costs for each part are calculated as follows: the cost of bringing passengers in advance (wtp) e • Add (In) to Cost Arc 1 (time advance only); the cost corresponding to advancing passengers and spatial shifting. Add to Cost Arc2 (time advance, spatial offset); Cost Arc3 remains unchanged (no time, spatial offset); adjust the cost corresponding to passenger spatial offset. Add to Cost Arc 4 (spatial offset only); the cost of delaying passengers. Add to Cost Arc 5 (time delay only); the cost corresponding to passenger delay and spatial offset. Add this to Cost Arc 6 (time delay, spatial offset). Time flexibility cost (time advance) is the advance time difference multiplied by the time flexibility cost factor (time advance); time flexibility cost (time delay) is the delay time difference multiplied by the time flexibility cost factor (time delay); spatial flexibility cost is the offset distance multiplied by the spatial flexibility cost factor.

[0201] Therefore, total flexibility cost Add the cost of spatial flexibility to the cost of time flexibility. That is:

[0202]

[0203] In the above formula, for the case of departing early, Different values ​​are assigned depending on the degree of advance notice. The value is 0 for departures that are delayed, while the opposite is true for delayed departures. For example, suppose the maximum allowable offset for a passenger is 2 spatiotemporal network intervals In, i.e. So The possible values ​​are (2,0), (1,0), (0,0), (0,1), and (0,2). Figure 9 In this scenario, the ideal departure node for passengers is 11. Therefore, the values ​​for the advance departure time and the delay departure time at node 1 are: For the case of time advancement: the calculation process is as follows: This represents the offset from the ideal time. For the case of time delay: the calculation process is as follows: This represents the offset from the ideal time. If the site has been offset, then (Regarding n and n′, see Algorithm 1, where n corresponds to Origin[1], and n′ corresponds to an element with an index of 2 or higher in the set Origin) is the distance between the stations before and after the offset, otherwise it is 0. Where n′=mod(i,N stop ), representing the spatiotemporal network node number i divided by the number of stations N. stop The remainder obtained afterward. Their relationship is as follows: Figure 9 As shown, assuming i = 12, then n′ = mod(i, N) stop) = mod(12, 10) = 2, which means the second station. And n is 1.

[0204] t i This is the time of node i, expressed in terms of time on the spatiotemporal network, and is a multiple of the spatiotemporal network interval In. Taking node 12 as an example, t 12 The calculation process is as follows:

[0205] The arc cost of the spatiotemporal networks of vehicle flow and passenger flow is respectively represented by the following piecewise functions:

[0206]

[0207]

[0208] 6. The setting of the boarding and alighting node set is specifically as follows:

[0209] For passengers, the time spent on the bus is also a part of passenger satisfaction. If the bus keeps making detours to pick up more passengers, the excessively long waiting time will disrupt passengers' plans. Therefore, our problem needs to consider the relationship between passenger waiting time and the journey time from the boarding point to the alighting point. For bus companies, serving as many passengers as possible with one bus can reduce costs and increase efficiency. Achieving this requires relaxing some passengers' arrival time windows, that is, allowing some passengers to delay their alighting time to serve more passengers. However, to protect passengers' interests and satisfaction, we must limit the maximum passenger waiting time t. iv We set t iv The journey time t for a vehicle from the boarding station to the alighting station. d Multiply by the delay factor For example, as shown in Figure 10(a), there are 3 stations. The shortest travel time from station 1 to station 2 is 15 minutes, from station 1 to station 3 is 20 minutes, and from station 2 to station 3 is 15 minutes. Suppose a group of passengers wants to depart from station 1 at 06:00 and reach station 3. As shown in Figure 10(b), if the vehicle travels directly from station 1 to station 3, the passengers will arrive at station 3 at 06:20. Another group of passengers wants to depart from station 2 at 06:15 and also reach station 3. If they arrive at station 3 at 06:20, the second group of passengers cannot be served. If we let... If the value is 1.5, the longest time a passenger can stay on the bus is 30 (20 × 1.5 = 30) minutes. Then, if the arrival time of the first batch of passengers is delayed by 10 minutes, the passengers can first depart from station 1 to station 2, and then go to station 3, thus serving two batches of passengers at the same time.

[0210] Extracting boarding and alighting nodes from the spatiotemporal network is a prerequisite for model input and a fundamental step in solving the problem. Below, we will discuss how to extract boarding and alighting nodes while providing flexibility.

[0211] (1) Set of boarding nodes

[0212] The set of boarding nodes is created according to the following formula (13). This represents the set-fetching function for the boarding node.

[0213]

[0214] It is the first The order data in the layered passenger flow spatiotemporal network is generated from passenger demand. It includes multiple passenger boarding stations (the first being the ideal boarding station), multiple passenger alighting stations (the first being the ideal alighting station), ideal boarding time, passenger quantity, and time flexibility parameters. (Integer, representing a multiple of the time interval in the spatiotemporal network, which can be determined by the bus company based on actual conditions; the time flexibility parameter indicates how much departure time offset passengers are allowed). N Stop It refers to the number of stations. It is the upper limit of service time (an integer, which is a multiple of the time interval of the spatiotemporal network).

[0215] Using order data and other input parameters, the boarding station number is first obtained. This number is then combined with time flexibility parameters to obtain the boarding spatiotemporal network node. For example, such as... Figure 11 As shown, the spatiotemporal network node numbers in the diagram increment by station number within the same row. After traversing all stations, they continue to increment in chronological order. For example, the first row (6:00) shows numbers from 1 to 10 (number of stations), and the second row (6:05) shows numbers from 11 to 20. (From order data...) The boarding station numbers are 1 and 2, and the boarding time is 2 (a multiple of the network time interval In, representing the actual time as 6:05, where In = 5 minutes). 1 =1, and N stop =10, By f 1 Given that time = 1 and boarding time is 2, passengers are allowed to depart one hour earlier or later, meaning departures at times 1 (6:00), 2 (6:05), and 3 (6:10) are all possible. Knowing that boarding station numbers are 1 and 2, the spatiotemporal network node numbers for boarding are (1, 2, 11, 12, 21, 22). These spatiotemporal network node numbers represent (Station 1 Time 1), (Station 2 Time 1), (Station 1 Time 2), (Station 2 Time 2), (Station 1 Time 3), and (Station 2 Time 3), respectively.

[0216] (2) Set of disembarkation nodes

[0217] In the model, by setting Make the set of passenger disembarkation nodes If it is fixed, then the function of delaying disembarkation can be realized to serve more orders.

[0218]

[0219] Equation (14) above represents the method for setting the set of disembarkation nodes. Wherein... This represents the set-fetching function for the disembarkation node. A matrix representing the direct travel time between representative stations.

[0220] Using order data, boarding time-space network nodes, and other input parameters, the drop-off station number Destination[i] is first obtained. For each drop-off station, the direct travel time from the boarding station to that drop-off station is considered. This time, plus the boarding time, equals the drop-off time, which is then converted to the earliest drop-off time-space network node. Considering the demand-response bus's need to accommodate more passengers and passenger satisfaction, a maximum on-board time is set, which is equal to the direct travel time. The time interval is multiplied by 1 (and rounded up to the nearest integer multiple of the network time interval). Adding the boarding time gives the disembarkation time. After conversion, this becomes the latest disembarkation time-space network node. Among all nodes between the earliest and latest disembarkation time-space network nodes, the node with disembarkation station number Destination[i] is a permitted disembarkation time-space network node. For example, such as... Figure 12 As shown, from order data The alighting station number is obtained as 3, and N is another one. Stop =10, Assumption From The boarding station is 2 and the time is 2. (The network time interval is a multiple of In, which here represents the actual time in 10 minutes). Therefore, the earliest drop-off time for passengers (direct) can be obtained as follows: Latest time for passengers to get off (considering) After delaying the drop-off time, the latest drop-off time is Given that the disembarkation station is 3, the spatiotemporal network node number for disembarkation is (33, 43). These spatiotemporal network node numbers represent (station 3 time 4) and (station 3 time 5), respectively.

[0221] Construct a demand-response bus scheduling optimization model

[0222] Based on the above introduction to the vehicle / passenger flow spatiotemporal network model, the model can be viewed as a special type of multi-goods network flow problem. The goal of the model is to exploit the spatiotemporal flexibility of passengers under stochastic travel times, while facilitating the flow of vehicles and passengers in the network and minimizing total cost.

[0223] Our objective is to minimize total cost. For a set of pre-determined passenger demands, the goal of the demand-response bus scheduling model is to calculate the optimal routes and timetables for demand-response buses to serve these demands. For passengers, if a passenger's demand can be met by the dispatch center's calculations, the passenger will be assigned to a bus and travel within the planned time. Otherwise, service is refused, which will affect passenger satisfaction. Therefore, in our model, to minimize the occurrence of refused passengers, we set a parameter I to represent the penalty for not serving a passenger order.

[0224] Objective function. Formula (15) minimizes the total cost, which includes the operating costs of the demand-response bus (driver costs and transportation costs), the travel costs of all passengers on the bus, and penalties for unserved passengers. We use minimizing the total cost as the optimization objective to calculate the model.

[0225]

[0226] Vehicle flow spatiotemporal network parking lot constraints. Formulas (16) and (17) indicate that the number of vehicles leaving the parking lot and returning to the parking lot is the same in each vehicle flow spatiotemporal network, thus ensuring that vehicles leaving the parking lot can return to the parking lot.

[0227]

[0228]

[0229] Traffic flow conservation constraint for nodes in the traffic flow spatiotemporal network. Formula (18) represents the traffic flow conservation of each node in the traffic flow spatiotemporal network. This constraint is to ensure the consistency of traffic flow when entering and leaving nodes in the traffic flow spatiotemporal network. That is, in the stations of the traffic flow spatiotemporal network, after the demand response bus arrives at this station from other stations, the next step is to continue to perform the task or return to the depot.

[0230]

[0231] The flow conservation constraint of supply (departure point) and demand (destination point) nodes in the passenger flow spatiotemporal network. Formulas (19) and (20) represent the flow conservation of each supply node and demand node in each passenger flow spatiotemporal network. This constraint is designed based on the problem to form a complete passenger travel chain, and a virtual "passenger depot" is introduced. It is the first A virtual "passenger parking lot" in the spatiotemporal network of passenger flow. Then it means in the first In the spatiotemporal network of passenger flow, passengers are transported from the virtual "passenger depot" to the departure point; Then it means in the first In the spatiotemporal network of passenger flow, passengers need to return to the virtual "passenger parking lot" after arriving at their destination.

[0232]

[0233]

[0234] Passenger flow spatiotemporal network node flow conservation constraint. Equation (21) represents the flow conservation of each node in each passenger flow spatiotemporal network. This constraint is to ensure the consistency of flow between inbound and outbound passenger flow spatiotemporal network nodes.

[0235]

[0236] Passenger Number Constraint. Formula (22) shows that the sum of passenger travel arcs multiplied by the number of passengers in the passenger flow spatiotemporal network should be less than or equal to the sum of the corresponding vehicle travel arcs multiplied by the vehicle capacity in all vehicle flow spatiotemporal networks. This constraint ensures that passengers can board vehicles, because for passengers to travel from node i to node j, there must be vehicles to transport them. For vehicles, only by traveling from node i to node j can they serve the corresponding demand of passengers. The set of all arcs in the vehicle (passenger) flow spatiotemporal network. In the diagram, arc ij represents the journey from node i to node j (whether it's a demand-response bus or a passenger). Arc (i,j) connects the demand-response bus and the passenger, ensuring that the passenger boards the demand-response bus to travel from node i to node j.

[0237]

[0238] Decision variable constraints. Formulas (23)-(26) stipulate that all decision variables in the vehicle flow spatiotemporal network and passenger flow spatiotemporal network are 0 or 1.

[0239]

[0240]

[0241]

[0242]

[0243] II. Detailed Implementation Analysis

[0244] describe

[0245] To demonstrate the correctness and applicability of the proposed model, we used the well-known Sioux Falls network for optimization computation. For example... Figure 13 As shown, the network has 24 nodes and 38 bidirectional road segments. The numbers on the road segments in the diagram represent the travel time (in minutes), using the average vehicle speed. This converts 30 km / h to distance. In this example, we consider a time range of 13 timestamps (from 8:00 to 10:00, with 10-minute intervals) (times will be represented by numbers below, e.g., 1 represents 8:00). This time interval is adjustable and can be modified as needed. Of course, theoretically, the higher the node density (i.e., the shorter the time interval), the more accurate the result, but the larger the problem size.

[0246] Table 5 shows the needs of all passengers. Passenger ID refers to a specific group of passengers. Passengers can choose two reserved pick-up points, with the first being the first ideal departure point and the second being the second ideal departure point. The second ideal departure point is optional. The same applies to reserved drop-off points. Travel time refers to the passenger's ideal pick-up time. For demonstration purposes, the passenger's time flexibility is set to 1 (i.e., the passenger's travel time can be 10 minutes earlier or later).

[0247] Table 5 Passenger Demand

[0248]

[0249] Taking the first batch of passengers as an example, the passengers' first ideal departure station is station 1, the second ideal departure station is station 2, the passengers' first ideal destination station is station 22, and the second ideal destination station is station 20. Then, based on time flexibility and spatial flexibility, the passengers' boarding time windows are (1,4), (1,5), (1,6), (2,4), (2,5), (2,6).

[0250] Random travel times and their corresponding probabilities are generated randomly; other parameters used are shown in Table 6. Considering the previously mentioned ρ... i,j ε i,j ζ i,j The operating hours may be divided into four periods, each lasting 30 minutes, depending on the time of day. These periods are: 8:00-8:30, 8:31-9:00, 9:01-9:30, and 9:31-10:00. Since the drivers' recovery effects are roughly the same, a ρ value is set. i,jand ζ i,j The value is 0.95 across all periods. However, the cumulative effect on passengers varies significantly across different periods, therefore ε is set... i,j The values ​​for the four periods are: 1.2, 1.2, 1.5, and 1.5.

[0251] Parameters used in the example in Table 6

[0252]

[0253]

[0254] result

[0255] By establishing a model and solving it using programming, we obtain the final result, which we will now analyze.

[0256] Figures 14(a)-14(b) The diagram illustrates vehicle scheduling with and without spatiotemporal flexibility, clearly demonstrating its significant impact on fleet size and scheduling. Here, spatiotemporal flexibility is defined as a model where both origin and destination have spatial flexibility, and time flexibility is 20 minutes. With spatiotemporal flexibility, bus operators can expand the spatial coverage of their fleet (19→17→16→8) to meet greater demand, reducing the fleet size from 4 vehicles to 3.

[0257] Figures 15(a)-15(c) This is a detailed parameter comparison. Considering spatial and temporal flexibility, the total vehicle and passenger costs for the entire transportation system are 2609.1 yuan, requiring 3 vehicles and having 3 unserved passenger orders. Without considering spatial and temporal flexibility, the total vehicle and passenger costs are similar at 2614.4 yuan, but require 4 vehicles and have 6 unserved passenger orders, resulting in more rejected orders and lower passenger satisfaction. Without considering spatial and temporal flexibility, more vehicles are needed but fewer orders are served, while considering it requires fewer vehicles, serves more orders, and slightly reduces vehicle and passenger costs. To a certain extent, this can be considered as stronger "aggregation" and higher vehicle passenger utilization. This not only saves costs but also brings better social benefits and improves passenger satisfaction with demand-response public transportation. This demonstrates the significant role of leveraging spatial and temporal flexibility.

[0258] Sensitivity analysis

[0259] ①Time flexibility

[0260] Time flexibility describes the limits to which passenger departure times can deviate, which is closely related to the bus company's operating model, so we conduct a sensitivity analysis on it here.

[0261] With no spatial flexibility set, the time flexibility parameter was adjusted from 0 to 1 and 2 respectively (i.e., the offset time was adjusted from 0 to 10 minutes and 20 minutes), and the results were as follows. Figures 16(a)-16(b) As shown in the figure, the number of service orders initially increases and then decreases with increasing time flexibility, but is generally higher than the initial level. Considering the fleet size, the possible reasons are as follows: When time flexibility increases from zero to 10 minutes, the expanded time window allows for more flexible passenger departure times, enabling more passengers to board, thus increasing the number of service orders. Compared to the initial situation, passengers are more "aggregated." When time flexibility further increases to 20 minutes, passenger departure times become even more flexible. From a system-optimal perspective, reducing one vehicle and one service order during optimization still results in a decrease in overall cost. Compared to the situation without time flexibility, fewer vehicles can serve more orders, resulting in higher "aggregation." This provides an insight for bus companies: when fleet size is limited, a suitable value can be found for operation. At this point, a smaller fleet size can complete more service orders, maximizing the passenger capacity of existing vehicles and improving service levels within limited costs.

[0262] ② Spatial flexibility

[0263] Whether passengers' departure and destination stations are allowed to deviate is reflected in spatial flexibility, which is just as important as temporal flexibility. Therefore, we conduct a sensitivity analysis on it here.

[0264] With no time flexibility set up, the scenarios were divided into four types by adjusting the spatial flexibility parameters: no spatial flexibility (Scenario 1), origin point spatial flexibility (Scenario 2), destination point spatial flexibility (Scenario 3), and origin and destination point spatial flexibility (Scenario 4). The results are shown in Figure 17(a). Figure 17(b) shows that the origin and destination spatial offset values ​​differ under different spatial flexibility modes. The cost with spatial offset is lower than the cost without spatial offset. While the fleet size remains consistent across all scenarios, the number of service orders is lowest with no spatial flexibility, highlighting the value of exploring spatial flexibility. Furthermore, the results also indicate that considering both origin and destination point spatial flexibility (Scenario 4) yields the highest overall benefit.

[0265] ③ Vehicle fixed costs

[0266] In bus company operations, operators are highly concerned about vehicle fixed costs, as they relate to fleet size and are a significant component of operating costs. Exploring the impact of these costs on different operational performance levels is therefore meaningful. Consequently, we conducted a sensitivity analysis of vehicle fixed costs, and the results are shown in Table 7.

[0267] The improvement in Table 7 refers to the difference between the results with and without spatiotemporal flexibility, calculated using the following formula: Where ω 无 ω represents the result of the absence of spatiotemporal flexibility. 有 This represents the results of the presence of spatiotemporal flexibility. Here, spatiotemporal flexibility refers to a pattern with spatial flexibility at origin and destination, and a time flexibility of 20 minutes. As vehicle fixed costs increase, total costs also increase. When spatiotemporal flexibility exists, total costs and fleet size can decrease while the number of service orders increases. With increasing vehicle fixed costs, the overall improvement in total costs and the number of service orders increases, while the improvement in fleet size decreases. When vehicle fixed costs are relatively low (200), although the improvement in the number of service orders is not significant (only 6.25%), the fleet size is significantly reduced (40%). When vehicle fixed costs increase to 400, without spatiotemporal flexibility, the number of service orders decreases by 4 with the same fleet size. However, with spatiotemporal flexibility, even with relatively high vehicle fixed costs (400), the service level (number of service orders) remains unchanged. This provides an insight that when vehicle fixed costs are relatively low, spatiotemporal flexibility can significantly reduce fleet size, thus reducing operating costs, while simultaneously improving service levels; when vehicle fixed costs are high, spatiotemporal flexibility can significantly improve service levels with the same fleet size.

[0268] Table 7 System performance indicators for different vehicle fixed costs with and without spatiotemporal flexibility.

[0269]

[0270]

[0271] ④ Penalties for passengers who do not receive service

[0272] Table 8 System performance indicators under different unserved passenger penalty costs with and without spatiotemporal flexibility.

[0273]

[0274] In our model, most parameters are based on actual values, while the penalty cost I for rejecting orders (unit penalty cost) is customized. Since vehicle and passenger costs are positive, the penalty cost I is used in the minimization problem to weigh additional vehicle costs, passenger costs, and user satisfaction to ensure a certain number of orders are served. In other words, without the penalty cost, no orders are served. We conducted experiments, and the results are shown in Table 8. As expected, total cost, fleet size, and the number of serviced orders increase with increasing unit penalty cost. When spatiotemporal flexibility exists, more orders can be served with a smaller fleet size (except when the unit penalty cost is 200), and the cost is also lower. With increasing unit penalty cost, the overall improvement in total cost and fleet size increases, while the improvement in the number of serviced orders decreases. When the unit penalty cost is 150, no orders are served; when the unit penalty cost increases to 350, the number of serviced orders is the same with and without spatiotemporal flexibility.

[0275] ④ Sensitivity analysis under the full order service model

[0276] The sensitivity analysis described the scenario of matching appropriate capacity to meet demand, in which demand can be traded. Next, we will explore how performance indicators such as capacity change under the same capacity conditions. Based on the model's characteristics, when the penalty for unserved passengers reaches a certain level, all orders will be served, and the capacity will be the same. We call the situation where all orders are served "no order rejection allowed." After multiple experiments, we found that a penalty of 2500 for unserved passengers allows for full service; therefore, this parameter is used for analysis.

[0277] Independent analysis of temporal and spatial flexibility

[0278] Table 9 System performance indicators under different time flexibility when order rejection is allowed and not allowed.

[0279]

[0280]

[0281] First, we evaluate the independent effect of time flexibility, i.e., a sensitivity analysis without considering spatial flexibility, and the results are shown in Table 9. Since introducing time and space flexibility can improve vehicle utilization, we introduce an indicator called the return on investment (ROI) to assess the system's resource usage. This indicator is calculated as the ratio between fleet size and the number of service orders. The first observation is that total cost and ROI decrease as time flexibility increases. Under different time flexibility levels, the time offset, total cost, and ROI for the mode allowing order rejection are significantly lower than those for the mode not allowing order rejection. This is because allowing order rejection allows the system to select "valuable" demands in a more economical way. The effect of time flexibility is more pronounced when order rejection is allowed.

[0282] When time flexibility increases from 0 to 10 minutes, the return on investment (ROI) decreases by approximately 5% for allowing order rejection and 0 for not allowing order rejection; when time flexibility increases from 10 minutes to 20 minutes, the ROI decreases by approximately 5% for allowing order rejection and 10% for not allowing order rejection. This indicates that as time flexibility increases, its incremental contribution to ROI also increases. Although the ROI can decrease by approximately 10% with a time flexibility of 20 minutes when order rejection is not allowed, it can decrease by approximately 5% with a time flexibility of only 10 minutes when order rejection is allowed, while there is no improvement when order rejection is not allowed. This suggests that when order rejection is allowed, the ROI can decrease more easily with lower time flexibility.

[0283] Table 10 illustrates the independent effects of spatial flexibility. As expected, greater spatial flexibility leads to lower total costs and return on investment (ROI). Under different levels of spatial flexibility, the allowable spatial offset for order rejection, total cost, and ROI are significantly lower than those without allowable rejection. The effects of origin flexibility (from scenario 1 to scenario 2) and destination flexibility (from scenario 1 to scenario 3) are not significantly different. However, when origin flexibility (from scenario 2 to scenario 4) or destination flexibility (from scenario 3 to scenario 4) is present, the incremental contribution of spatial flexibility to ROI becomes very small.

[0284] Table 10 System performance indicators under different spatial flexibility when order rejection is allowed and not allowed.

[0285]

[0286] The effect of time flexibility

[0287] As we can see from the previous section, in this example, the independent effect of time flexibility is more significant from the perspective of cost and input-output ratio. To explore the effect of time flexibility, we studied its impact under conditions of origin flexibility and destination flexibility (full spatial flexibility).

[0288] The results are as follows Figures 18(a)-18(b) As shown in Figure 18(a), the value of the top dashed line represents the total cost without spatial and temporal flexibility and without order rejection (base case). With full spatial flexibility, the total cost can be reduced by 7.94%. Based on full spatial flexibility, when the time flexibility increases from 0 to 10 minutes, the total cost is reduced by 14.15% compared to the base case; when the time flexibility increases from 10 minutes to 20 minutes, the total cost decreases by 14.95% compared to the base case. According to previous results, when the time flexibility is 10 minutes and there is no spatial flexibility, the total cost is reduced by only 0.36%. However, when the time flexibility is 10 minutes and there is spatial flexibility, the total cost can be reduced by 14.15%, which is even higher than the result when the time flexibility is 20 minutes but there is no spatial flexibility.

[0289] As shown in Figure 18(b), with increasing time flexibility, a smaller fleet size leads to a larger spatial offset between departure and destination, while the departure time offset initially increases and then decreases. This is because the additional cost of time flexibility (20 minutes) outweighs the cost of spatial flexibility. The results indicate that the independent contributions of time flexibility and spatial flexibility are limited, but their combined effect is better. Furthermore, under certain conditions, parameters can be selected as needed to minimize costs.

[0290] The above embodiments are preferred embodiments of the present invention, but the embodiments of the present invention are not limited to the embodiments described above. Any changes, modifications, substitutions, combinations, or simplifications made without departing from the spirit and principle of the present invention shall be considered equivalent substitutions and shall be included within the protection scope of the present invention.

Claims

1. A demand-response bus scheduling method considering stochastic road networks and passenger spatiotemporal flexibility, characterized in that, include: Constructing a spatiotemporal network for vehicle flow and a spatiotemporal network for passenger flow; Passenger flexibility verification; The cost of setting up arcs in the spatiotemporal networks of vehicle flow and passenger flow; Set the set of boarding and alighting nodes; Construct a demand-response bus scheduling optimization model, solve it, and obtain the demand-response bus scheduling optimization scheme. The passenger flexibility verification is reflected by the amount that passengers are willing to pay for different levels of flexibility in time and space; The method for setting the cost of arcs in the vehicle flow spatiotemporal network and passenger flow spatiotemporal network is as follows: The arc cost of the spatiotemporal networks of vehicle flow and passenger flow is respectively represented by the following piecewise functions: in: The cost per unit of time for the driver; The cost per unit of time a passenger spends on the vehicle; The variable cost per kilometer of vehicle transportation; The fixed cost of dispatching each vehicle; For train number The driving distance; and They are nodes and nodes Time; The time interval of the spatiotemporal network; For the first Train numbers in the spatiotemporal network of layered traffic flow Planned travel time The total cost of punishment; It is the first Train numbers in the spatiotemporal network diagram of passenger flow Planned travel time Total penalty cost, For total flexibility cost, For the arc cost of the vehicle flow spatiotemporal network, The arc cost of the passenger flow spatiotemporal network; Total Flexibility Cost Add space flexibility costs to time flexibility costs: For cases where the time is brought forward: This represents the advance departure time corresponding to node i. For cases where time is delayed: It is a node The corresponding delay in departure time, If the site has been offset, This indicates the distance between a passenger's ideal pick-up point and alternative pick-up points. , These are the unit time flexibility costs when the time is brought forward, the unit time flexibility costs when the time is delayed, and the unit space flexibility costs. The method for constructing a demand-response bus scheduling optimization model is as follows: Define an objective function that makes the total operating cost... Minimize: in, Indicates the first Arc in Layered Vehicle Flow Network the flow, Indicates the first In the arc of the passenger flow network the flow, Indicates if the first A value of 1 indicates that a passenger associated with the passenger flow network has been served; otherwise, a value of 0. The number of passengers boarding in the passenger flow network. The index representing the passenger flow network, i.e., the first... Layered passenger flow network This indicates a penalty for failing to serve a passenger order. The index of the traffic flow network, i.e., the first... Layered vehicle flow network Indicates the first The set of all arcs in the layered traffic flow network. Indicates the first The set of all arcs in the passenger flow network; Passenger flow spatiotemporal network supply and demand node flow conservation constraint: ； ； in, Indicates the first Virtual parking lots in a multi-level passenger flow network For the set of boarding nodes, This is the set of disembarkation nodes; Passenger flow spatiotemporal network node flow conservation constraint: ； in, Indicates the first The set of all nodes in the passenger flow network; The set of boarding and alighting nodes is extracted from the passenger flow spatiotemporal network, specifically as follows: Set of boarding nodes : ； in This represents the set-fetching function for the boarding node. It is the first The order data in the spatiotemporal network of passenger flow is generated from passenger demand. It refers to the number of stations. This is the maximum service time. Set of disembarkation nodes : ； in This represents the set-fetching function for the disembarkation node. A matrix representing direct travel time between representative stations. This represents the delay factor.

2. The demand response bus dispatching method according to claim 1, characterized in that, include: The traffic flow spatiotemporal network corresponds to a demand-response bus and reflects the driving status of each bus; There are two axes: the horizontal axis represents the spatial dimension, indicating spatial location, specifically a station; the vertical axis represents the time dimension, indicating the time corresponding to a certain event. Nodes in the traffic flow spatiotemporal network have temporal and spatial attributes, representing the location of a bus at a certain time. In the spatiotemporal network of traffic flow, an arc represents the trajectory of a vehicle.

3. The demand response bus dispatching method according to claim 2, characterized in that, The arcs in the spatiotemporal network of the traffic flow include four types, namely: Random vehicle travel arc: represents the journey of a bus from one station to another, which corresponds to a journey time. The corresponding arc cost is the variable cost of the vehicle performing the transportation task plus the penalty cost that reflects the possible early or late arrival in actual operation rather than the trip plan. Vehicle waiting arc: represents the time a vehicle spends waiting at a certain station during a certain period. The corresponding arc cost is set as the vehicle's waiting time multiplied by the driver's cost per unit of time. Vehicle departure arc: This arc connects the depot and the station. The departure end is the depot, and the destination end is the passenger boarding station. The corresponding arc cost represents the operating cost of arranging vehicle departure. Vehicle return arc: This refers to the vehicle returning to the depot after serving the last passenger and reaching the last drop-off point. The arc cost of the vehicle return arc is set to 0.

4. The demand response bus dispatching method according to claim 1, characterized in that, The passenger flow spatiotemporal network describes passenger flow patterns, with each layer of the passenger flow spatiotemporal network corresponding to a passenger demand with a start-destination-time. The passenger flow spatiotemporal network diagram has two axes. The vertical axis represents the time dimension, that is, the time corresponding to a certain event. The horizontal axis represents spatial dimensions, that is, spatial location; In the spatiotemporal network of passenger flow, nodes and arcs are set up. Nodes have time and space attributes, and arcs represent the movement of passengers. Arcs correspond to the starting node and the destination node. The duration of the passenger flow spatiotemporal network represents the travel time window allowed by the relevant origin-destination-time requirements.

5. The demand response bus dispatching method according to claim 4, characterized in that, The passenger flow spatiotemporal network includes four types of arcs, namely: Passenger random travel arc: The passenger random travel arc represents the passenger journey from one station to another, which corresponds to a journey time. The number of random travel arcs in each passenger flow spatiotemporal network corresponds to the number of vehicle random travel arcs in the vehicle flow spatiotemporal network. Its arc cost is specifically the variable cost of serving passenger travel plus the penalty cost that reflects the possible early or late arrival in actual operation rather than the trip plan. Passenger waiting arc: This represents the time a passenger spends waiting at a station during a certain period. The cost of the passenger waiting arc is set as the passenger's waiting time multiplied by the cost per unit time the passenger spends on the train. Passenger Virtual Departure Arc: Connects the virtual depot and the station. The departure point is the virtual depot, and the destination point is the passenger boarding station. Its corresponding arc cost is set to 0. Passenger Virtual Return Arc: This represents the return of a passenger to the virtual depot after arriving at their destination station, and its corresponding arc cost is 0.

6. The demand response bus dispatching method according to claim 1, characterized in that, The order data includes multiple passenger pick-up points, multiple passenger drop-off points, ideal pick-up time, number of passengers, and time flexibility parameters. Among multiple passenger boarding points, the first boarding point is the ideal boarding station, and among multiple passenger alighting points, the first alighting point is the ideal alighting station.

7. The demand response bus dispatching method according to claim 1, characterized in that, The constructed model is a mixed-integer linear programming model.

Images